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We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular perturbation theory to explore the dependence of the geometry…
Two-stroke relaxation oscillations consist of two distinct phases per cycle - one slow and one fast - which distinguishes them from the well-known van der Pol-type 'four-stroke' relaxation oscillations. This type of oscillation can be found…
Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of 'stationary' multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast…
Phenomena in chemistry, biology and neuroscience are often modelled using ordinary differential equations (ODEs) in which the right-hand-side is comprised of terms which correspond to individual 'processes' or 'fluxes'. Frequently, these…
We present a novel and global three-dimensional reduction of a non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations [J. Rubin and M. Wechselberger, Giant squid--hidden canard: the 3D geometry of the Hodgkin-Huxley…
In this paper, we describe a novel type of relaxation oscillations occurring in a model of substrate-depletion oscillators. Using geometric singular perturbation theory, with blow-up as a key technical tool, we show that the oscillations in…
In the world of technology, one of the most important forms of friction is that of rolling friction. Yet it is one of the least studied of all the known forms of energy dissipation. In the present experiments we investigate the oscillatory…
We study non oscillating bifurcations of non homogeneous steady states of the Vlasov equation, a situation occurring in galactic models, or for Bernstein-Greene-Kruskal modes in plasma physics. We show that resonances are strongly…
Performing molecular dynamics simulations for all-atom models, we characterize the conformational and structural relaxations of poly(ethylene oxide) and poly(propylene oxide) melts. The temperature dependence of these relaxation processes…
We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method.…
In this paper, Van der pol equation has been analyzed for stability and bifurcation phenomena with and without forcing component. Analytical solution of the Van der pol equation using Method of Multiple Scales (MMS) is compared with…
The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely…
We analyze the convergence of a perturbed circular interface for the two-phase Mullins-Sekerka evolution in flat two-dimensional space. Our method is based on the gradient flow structure of the evolution and captures two distinct regimes of…
A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary…
We study a two-variable dynamical system modeling pH oscillations in the urea-urease reaction within giant lipid vesicles -- a problem that intrinsically contains multiple, well-separated timescales. Building on an existing, deterministic…
Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations and multistability. The results…
Simple conditions have been developed in [Zhang, Wahl and Yu, SIAM Rev. 2014; Yu and Wang, Math. Biosci. Eng. 2019], which are used to identify the existence of slow-fast relaxation oscillations that appear in differential systems, where…
In this work we study the Brusselator - a prototypical model for chemical oscillations - under the assumption that the bifurcation parameter is of order $O(1/\epsilon)$ for positive $\epsilon\ll 1$. The dynamics of this mathematical model…
Origami metamaterial design enables drastic qualitative changes in the response properties of a thin sheet via the addition of a repeating pattern of folds based around a rigid folding motion. Known also as a mechanism, this folding motion…
In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in…